Operations Research Questions

footprint University emailprotected Faculty on the job(p)s written document Lubin schooling of Business 11-1-1999 The riddle of Linear computing railcar programing Explained Second Edition goose Yurkiewicz charge per unit University Follow this and additional works at http//digitalcommons. pace. edu/lubinfaculty_workingpapers Recommended Citation Yurkiewicz, bastard, The Mystery of Linear scheduling Explained Second Edition (1999). Faculty Working Papers. Paper 21. http//digitalcommons. pace. edu/lubinfaculty_workingpapers/21 This Article is brought to you for forgo and open access by the Lubin tame of Business at emailprotectedIt has been accepted for inclusion body in Faculty Working Papers by an authorized administrator of emailprotected For to a greater extent information, please contact emailprotected edu. WORKING round legislate No. 191 November 1999 The Mystery of Linear program Explained Second Edition by Jack Yurkewicz, Ph. D. Professor of Management perce ption and theater director of the Advanced Professional security department weapons platform Lubin School of Business ill- ingestion University THE whodunit OF LINEAR PROGRAMMING EXPLAINEDSECOND EDITION As Recounted by John H. Watson, M. D.Edited by Jack Yurkiewicz, Ph. D. Jack Yurkiewicz is Professor of Management Science and Director of the Advanced Professional Certificate Program at the Lubin School of Business, Pace University. Introduction INTRODUCTION I hadnt seen my mature friend and companion, Sir Sherlock Holmes (he had been knighted by the queer and insisted that this appellation be customd) in more than(prenominal) or lesswhat(prenominal) months. Feeling that I had slighted him, I rang him up and told him I would be stopping by the next twenty- quaternion hours to see how his bran-new bank line of descent reckon was progressing.Holmes was sincerely refulgent to hear my voice and urged that we entirelyow lunch to breedher. While he hinted that his f irm, of which he was the doctor employee, was doing well, he did indeed wish to demonstrate a business proposition with me. needless to feel out, I was intrigued. I wont b some other to bore you, dear reader, with some of our mundane conversation the next day save to swear that I lay down Holmes to be moodier than ever. The great detective had great ago become bored with abomination solving, claiming there was little ch all in all toldenge in solving numerous of the violent crimes that seemed so prevalent in our society.The firm that he started provided consulting service to corporations, in which he would use his mental abilities to help managers solve their business-related bothers. Holmes beg offed to me that he enjoyed the challenge of his new endeavors weighed downly was decision it increasingly difficult to converse with these business theatrical roles, as he phoneed them. He b bely didnt brace the effort to explain his findings to them, for he launch them curiously dull. Holmes went on to imagine that he had no doubt that this event was the cause of the coun assays lack of success in the international business environment.I didnt requirement to say it, simply I thought to myself that the fault might non be precisely when with the business managers he was dealing with, alone partially with him, for I myself frequently found him short-tempered and impatient, even during his ordinary conversations with me. In any case, his proposition to me was simple. Holmes Listen, Watson, I safe crappert bear the thought of explaining to these so- scrubed leading how they should be running their firms. They come to me with some specific problem, usually ill-defined, and whence gestate me to come up with a result.I subscribe to found that arrive atting the solution is jolly trivial, but I get to the dickens of a time trying to explain it to them You take a crap a solid extend on your shoulders, old fellow. Suppose you became my pa rtner. Your only job would be to explain to them the solution that I propose, for I fear that my patience is not as dour as yours is. Watson The thought intrigues me, for I have been in the alto acquireher back on my practice late and do find myself available with more free time. What procedures do you use to get your results?Holmes Mostly a branch of math battle cryed operations research, which incidentally has its roots in this country during the Second World War. I collect the data, enter it on a computing instrument I have in my study, and in al closely an instant, I get the results. Watson Well, I am afraid that crying it, Holmes. I dont have sex the first matter slightly operations research and accordingly could not possibly explain it to others. I did try to let on the subject some(prenominal) long time back. I bought a college standard on the subject and spent some(prenominal) frustrating days trying to learn one-dimensional programming.The subject seemed u nderstandable enough, but I became hopelessly 1 The Mystery of Linear Program Explained lost on the mechanics of the simplex algorithm. I was working with equations and performing what the author called elementary speech operations. Believe me, there was nothing elementary nigh it. After leash days I threw the book down in disgust and came to the conclusion that only masochists or the temporarily insane would ever try to fathom such a subject. Holmes I fear you overestimate the subjects complexity, but the distributor point is irrelevant.No one really does these things by overstep anymore, and so it was effective your poor mint to buy a book that was obsolete. whole is instantaneously done with the aid of the computer. In fact, I am using a program called problem solver, which is a part of outmatch which is unusually easy to use and compelling. I allow explain the computer fruit to you, a unyielding with its ramifications, and then you in spell will explain it to the manager in question. It really is quite simple, and if I may say, also lucrative. The more Holmes spoke, the more provoke I became.His power of persuasion was truly formidable, and in the end, against much misgiving on my part, I agreed to enter into this venture on a trial basis. Holmes was squelched with my decision and proposed that we start spryly with a problem that he had completed the other day that he was due to root on the following week. HOLMES DESCRIBES THE CASE Holmes The Maximus calculating machine Company (MCC) has four basic computers it sells to students and polished business people. The first, called the newbie, is a basic, no-frills computer.It has most of the amenities that a new substance ab exploiter or a buyer on a strict budget could want, including CDROM with sound, an entry-level processor, a small warm disk, a modem for meshwork access, and a 15-inch monitor. The second modelling, called the Midrange, is for more demanding users. This model off ers a faster processor, larger hard disk, more RAM, a DVD player, and a 17-inch monitor. The third model, the Super, provides just about all the computing power a user could want. It offers even more RAM, a precise fast processor, a large hard disk, a DVD player with hardw ar decoder, and a 17-inch monitor.All but the most demanding users would be very(prenominal) happy with the Super. However, for those who want the very outmatch, the go with offers the original which offers a progressive processor, a huge hard disk, the best multimedia bringage (the latest coevals DVD ROM with a five-piece speaker system system), a CD re-writable drive, a 19inch monitor, etc.tera Thus, while the company offers only four models, it feels there is enough flexibility to cover most of the quarry computer audience. It is a small start-up company and management agnises it has to get by against the heavily entrenched products from Dell, IBM, Compaq, Hewlett-Packard, and Gateway.The company doctrine is to ship computers with brandknown components and offer superior service, all at a cost to consumers that is impose than the competitions. Watson Well, Holmes, I believe I know about the philosophy of the company and its immediate goals. Can you please range me more about the four models of computers we offer? 2 How the Operation Works Holmes As I said, Maximus delivers four models with different levels of features, all with the equivalent node support. They believe that whatever the mundanity of the machine, ruleer support should be top notch and not deteriorate with the price.In the long run, that will generate customer unsloped will and promote their growth in the market. Watson I agree whole-heartedly. Tell me a bit more about the computers themselves. HOW THE movement WORKS Holmes As you probably know, the computer industry is currently very competitive, and profits margins are low for distributively model. The fire profit on a Starter is $50, for a Midr ange it is $120, for a Super it is $250, and for an Extreme it is $300. These figures already create into cover material, labor, depreciation, taxes, shipping, etc. In other words, these are the bring in profits to the company for each computer sold.Watson Eventually we mint delve into how these determine are determined and perhaps how we locoweed increase them, but for now, lets assume that they are sacrosanct. What else can you tell me about the operation? Holmes Management has, what they call three operations that make a computer. They call the first operation manufacture. This includes victorious the customers phone call and determining which computer and options he or she wants and getting that information to the management of the end product staff. They in turn will get the necessary components and make them available to the workers on the throng line.The second operation is referred to as assembly, where the workers on the assembly line actually put together the com puter, according the specifications of the customers order. These are skilled workers who take pride in their work, and even though they work on an assembly line, they do not have an assembly line mentality, for they believe they are making custom products for specific customers. The last operation is called inspection. hither we install the software, run various symptomatic tests, and generally check out and pack up the computer before we give it to the customer. Watson I see. operations?Do you have approximate time figures on how long it takes to do these Holmes It is a function of the computer. However, they have been doing this for some time now and so the time value do not vary all that much from machine to machine. Thus, for instance, for a Starter, we can assume 0. 1 hours for manufacture, 0. 2 hours for assembly, and 0. 1 hours for inspection. Our units are forever and a day in hours,I hope that doesnt sidestep you. Watson Not at all. Actually, Im used to minutes and so I am making mental transformations, but since the company units are always measured in hours, I will use those also.Tell me the alike(p) value for the other computers. 3 The Mystery of Linear Programming Explained Holmes For a Midrange, we use 0. 2 hours to manufacture, 0. 5 hours to assemble, and 0. 2 hours for inspection. Each Super requires 0. 7 hours to manufacture, 0. 25 hours to assemble, and 0. 3 hours inspection and testing. Finally, the Extreme gets 0. 8 hours to manufacture, 0. 2 hours to assemble, and 0. 5 hours for inspection. As I said, these amount really dont vary all that much from machine to machine, but as you can see, they do differ from model to model.Watson Very well. What about your resources? How many people, or should I say, how many people-hours are available to do those three operations? Holmes I agree with you that we should prattle in terms of people-hours. I have been using the term man-hours for many years and if I lapse into that gender-specific ter m, please exculpate me. To avoid offending you, I will just use the word hours from now on, but I hope you know I mean people-hours when I say it. Watson Have no fear on my account, Holmes. My sensibilities will not be meet if you use the older term man-hours. What numbers do you have? Holmes On a daily basis, management informed me that the company has 250 hours available for manufacture, 350 hours available for assembly, and one hundred fifty hours to do the inspection and testing. Watson I presume that with all this information, we can proceed to model the problem as a bilinear program. Holmes Indeed, Watson. As I intimated earlier, we can solve linear programs with exceed. surmount comes with an add-on package called solver that is easy to use and yet powerful enough to solve most numeric programming problems.All we have to do is make a spreadsheet model of the problem and Solver will do the rest. Watson Well, that certainly is approximate news. I use outmatch and I hav e made many models. scarcely how do you make a linear program model in Excel? MODELING THE PROBLEM IN stick out Holmes Have a look at the Excel spreadsheet model here on my computer, Watson. It is imperative that you first learn how to model a problem in a spreadsheet. Once you master this skill, we can then proceed to discuss how Solver can be used to get the dish up for us.Cells B1, C1, D1, and E1 give the labels of our computers, which in linear programming terminology are called decision variables. We want the determine of these variables to appear in mobile phones B2, C2, D2, and E2. Excels Solver will call these jail cells the changing cells. We put the per-unit profit of each computer into cells B4 through and through D4. These numbers are traditionally called the objective function coefficients. We essential enter a formula into cell I2 (which we labeled as the loot in cell I1) that will give the discharge profit for all the computers made. 4 Modeling the Program in ExcelA 1 2 3 4 5 6 7 8 B Starter 0 50 0. 1 0. 2 0. 1 C Midrange 0 120 0. 2 0. 5 0. 2 D Super 0 250 0. 7 0. 25 0. 3 E Extreme 0 300 F G H I Profit 0 max manufacture assembly inspection Available 0. 8 250 0. 2 350 0. 5 150 Used slack off 0 250 0 350 0 150 Watson I know how to do that. We would instance in cell I2 the formula =B2*B4 + C2*C4 + D2*D4 + E2*E4 Algebraically, we are saying, with this formula 50(Starters) + 120(Midrange) + 250(Super) + 300(Extreme) Of course, the value in cell I2 is zero because we have zero values for the number of computers made in cells B2 through E2.Holmes Well done, Watson You should know that there is a shortcut to this rather tedious formula. Excel has the built-in function, SUMPRODUCT, which will save us much typing. That is, in cell I2 we can simply type =SUMPRODUCT(B2E2,B4E4) That says multiply the values in cells B2 through E2 by the corresponding values in the cells B4 through E4 respectively, and then add up the results. That gives us the si milar result as your formula. However, it is easier to input, since we need just type in the =sumproduct, highlight the ranges, and type the parentheses.The result is our total profit, which Solver calls the target cell. Watson I suspect that we will be using that sumproduct function again. Tell me about the other items I see in your spreadsheet. Holmes Our model is incomplete. We need to specify the reserves. These account for the technological, economic, or other limitations of the system, and restrict the values of the decision variables to some feasible set. We have three constraints in our problem manufacturing, assembly, and inspection. Our manufacturing constraint is 0. 1(Starter) + 0. 2(Midrange) + 0. 7(Super) + 0. 8(Extreme)